In general, apparatuses, computer programs, and methods for calculating one or more scattering parameters of a linear network are known. A N-port linear network has many equivalent representations. Two of them are the impedance matrix and the scattering matrix. The first one uses voltage and currents, as illustrated at the linear network in FIG. 8A, and can be more useful for calculations, the latter uses linear combinations of voltage and current at each port, as illustrated at the linear network in FIG. 8B, and is directly measurable, for example at high frequency.
Impedance and scattering parameters of the N-port linear network are often given in a N×N matrix referred to as Z-matrix and S-matrix, or even [Z] and [S], respectively. The matrices comprise the coefficients zhk and shk, respectively, which are complex functions of the frequency. The index h refer to the rows of the matrices and the index k refers to the columns of the matrices. Thus, for example, zhk is the Z-parameter found in the h-th row and the k-th column of the Z-matrix. For each network representation of the N-port linear network, there are three equivalent ways to write the system of N equations with N unknowns. Those are explicit equations (eq. 1 a, 1 b), compact form (eq. 2a, 2b), and matrix form (eq. 3a, 3b) of the impedance and scattering parameters. Thus, the values for ak and bk, which represent an incident wave and a reflected wave at a port k of the network respectively, follow from the equations shown in eq. 4, wherein R0 is the reference impedance, or normalization impedance of the scattering parameters.
                    Explicit        ⁢                                  ⁢        equations        ⁢                                  ⁢                  of          ⁢                                          [          Z          ]                                                                    {                                                                              V                  1                                =                                                                            z                      11                                        ⁢                                          I                      1                                                        +                                                            z                      12                                        ⁢                                          I                      2                                                        +                  …                  +                                                            z                                              1                        ⁢                        N                                                              ⁢                                          I                      N                                                                                                                                                                V                  2                                =                                                                            z                      21                                        ⁢                                          I                      1                                                        +                                                            z                      22                                        ⁢                                          I                      2                                                        +                  …                  +                                                            z                                              2                        ⁢                        N                                                              ⁢                                          I                      N                                                                                                                              …                                                                                            V                  N                                =                                                                            z                                              N                        ⁢                                                                                                  ⁢                        1                                                              ⁢                                          I                      1                                                        +                                                            z                                              N                        ⁢                                                                                                  ⁢                        2                                                              ⁢                                          I                      2                                                        +                  …                  +                                                            z                      NN                                        ⁢                                          I                      N                                                                                                                              (                              eq            .                                                  ⁢            1                    ⁢          a                )                                Explicit        ⁢                                  ⁢        equations        ⁢                                  ⁢                  of          ⁢                                          [          S          ]                                                                    {                                                                              b                  1                                =                                                                            s                                              1                        ⁢                        N                                                              ⁢                                          a                      1                                                        +                                                            s                                              1                        ⁢                        N                                                              ⁢                                          a                      2                                                        +                  …                  +                                                            s                                              1                        ⁢                        N                                                              ⁢                                          a                      N                                                                                                                                                                b                  2                                =                                                                            s                      21                                        ⁢                                          a                      1                                                        +                                                            s                                              2                        ⁢                        N                                                              ⁢                                          a                      2                                                        +                  …                  +                                                            s                                              2                        ⁢                        N                                                              ⁢                                          a                      N                                                                                                                              …                                                                                            b                  N                                =                                                                            s                                              N                        ⁢                                                                                                  ⁢                        1                                                              ⁢                                          a                      1                                                        +                                                            s                                              N                        ⁢                                                                                                  ⁢                        2                                                              ⁢                                          a                      2                                                        +                  …                  +                                                            s                      NN                                        ⁢                                          a                      N                                                                                                                              (                              eq            .                                                  ⁢            1                    ⁢          b                )                                Compact        ⁢                                  ⁢        form        ⁢                                  ⁢                  of          ⁢                                          [          Z          ]                                                                                          V            k                    =                                    ∑                              h                =                1                            N                        ⁢                                          z                kh                            ⁢                              I                h                                                    ,                  (                      h            ,                          k              =              1                        ,                          …              ⁢                                                          ⁢              N                                )                                    (                              eq            .                                                  ⁢            2                    ⁢          a                )                                Compact        ⁢                                  ⁢        form        ⁢                                  ⁢                  of          ⁢                                          [          S          ]                                                                                          b            k                    =                                    ∑                              h                =                1                            N                        ⁢                                          s                kh                            ⁢                              a                h                                                    ,                  (                      h            ,                          k              =              1                        ,                          …              ⁢                                                          ⁢              N                                )                                    (                              eq            .                                                  ⁢            2                    ⁢          b                )                                Matrix        ⁢                                  ⁢        form        ⁢                                  ⁢                  of          ⁢                                          [          Z          ]                                                                              [                                                                      V                  1                                                                                                      V                  2                                                                                    ⋮                                                                                      V                  N                                                              ]                =                              [                                                                                z                    11                                                                                        z                    12                                                                    …                                                                      z                                          1                      ⁢                      N                                                                                                                                        z                    21                                                                                        z                    22                                                                    …                                                                      z                                          2                      ⁢                      N                                                                                                                    …                                                  …                                                  ⋱                                                  ⋮                                                                                                  z                                          N                      ⁢                                                                                          ⁢                      1                                                                                                            z                                          N                      ⁢                                                                                          ⁢                      2                                                                                        …                                                                      z                    NN                                                                        ]                    ·                      [                                                                                I                    1                                                                                                                    I                    2                                                                                                ⋮                                                                                                  I                    N                                                                        ]                                              (                              eq            .                                                  ⁢            3                    ⁢          a                )                                Matrix        ⁢                                  ⁢        form        ⁢                                  ⁢                  of          ⁢                                          [          S          ]                                                                              [                                                                      b                  1                                                                                                      b                  2                                                                                    ⋮                                                                                      b                  N                                                              ]                =                              [                                                                                s                    11                                                                                        s                    12                                                                    …                                                                      s                                          1                      ⁢                      N                                                                                                                                        s                    21                                                                                        s                    22                                                                    …                                                                      s                                          2                      ⁢                      N                                                                                                                    …                                                  …                                                  ⋱                                                  ⋮                                                                                                  s                                          N                      ⁢                                                                                          ⁢                      1                                                                                                            s                                          N                      ⁢                                                                                          ⁢                      2                                                                                        …                                                                      s                    NN                                                                        ]                    ·                      [                                                                                a                    1                                                                                                                    a                    2                                                                                                ⋮                                                                                                  a                    N                                                                        ]                                              (                              eq            .                                                  ⁢            3                    ⁢          b                )                                                      a            k                    =                                    V              k                        +                                          R                0                            ⁢                              I                K                                                    ,                              b            k                    =                                    V              k                        -                                          R                0                            ⁢                              I                K                                                    ,                  (                                    k              =              1                        ,                          …              ⁢                                                          ⁢              N                                )                                    (                  eq          .                                          ⁢          4                )            
Relations between voltage (Vk) and current (Ik) and incident wave (ak) and reflected wave at a port k of the linear network (electrical response/reflected wave, bk)
From the relations between voltage and current and from incident wave and reflected wave, a reflected wave also being called an electrical response, respectively, it is possible to derive conversion formulae for converting [S] to [Z] and vice versa (see eq. 5 for [S] to [Z] & eq. 6 for [Z] to [S]).[Z]={[E]−[S]}−1E]+[S]}R0  (eq. 5)                Conversion formula for [S] to [Z][S]={[Z]−[E]R0Z]+[Z]R0}−1  (eq. 6)        Conversion formula for [Z] to [S]        
In the conversion formulae, [E] is the N×N unit matrix: the elements on the main diagonal (row index=column index) equal to one, all the remaining elements are zero, as shown in eq. 7.
                    N        ×        N        ⁢                                  ⁢        unit        ⁢                                  ⁢                  matrix          ⁢                                          [          E          ]                                                                              [          E          ]                =                  [                                                    1                                            0                                            …                                            0                                                                    0                                            1                                            …                                            0                                                                    …                                            …                                            ⋱                                            ⋮                                                                    0                                            0                                            …                                            1                                              ]                                    (                  eq          .                                          ⁢          7                )            
As shown in eq. 8, which follows from eq. 2a, the element on the h-th row and k-th column of the impedance matrix is, with regards to its physical meaning, the ratio between the voltage V on the port k and the current I injected on the port h, assuming that no current flows in the remaining ports. Thus, all the ports, excluding h may be left open circuit to determine the respective impedance parameters.
                    Definition        ⁢                                  ⁢        of        ⁢                                  ⁢        impedance        ⁢                                  ⁢        parameter        ⁢                                  ⁢                  z          kh                                                                                                                                          V                  k                                =                                                      ∑                                          h                      =                      1                                        N                                    ⁢                                                            z                      kh                                        ⁢                                          I                      h                                                                                  ,                              (                                  h                  ,                                      k                    =                    1                                    ,                                      …                    ⁢                                                                                  ⁢                    N                                                  )                                                                          ⇓                                                                                                                    z                    kh                                    =                                                            V                      k                                                              I                      h                                                                                                  ,                              (                                                                            I                      m                                        =                    0                                    ,                                      ∀                                          m                      ≠                      h                                                                      )                                                                        (                  eq          .                                          ⁢          8                )            
Furthermore, as shown in eq. 9, which follows from eq. 2b, the element on the h-th row and k-th column of the scattering matrix is, with regards to its physical meaning, the ratio between the wave b reflected by the port k and the wave a incident on port h, assuming that no other incident wave is applied. All the ports, excluding h, may be terminated by a resistance at or at least close to R0 in order to determine the respective scattering parameters of port h.
                    Definition        ⁢                                  ⁢        of        ⁢                                  ⁢        scattering        ⁢                                  ⁢        parameter        ⁢                                  ⁢                  s          kh                                                                                                                                          b                  k                                =                                                      ∑                                          h                      =                      1                                        N                                    ⁢                                                            s                      kh                                        ⁢                                          a                      h                                                                                  ,                              (                                  h                  ,                                      k                    =                    1                                    ,                                      …                    ⁢                                                                                  ⁢                    N                                                  )                                                                          ⇓                                                                                                                    s                    kh                                    =                                                            b                      k                                                              a                      h                                                                                                  ,                              (                                                                            a                      m                                        =                    0                                    ,                                      ∀                                          m                      ≠                      h                                                                      )                                                                        (                  eq          .                                          ⁢          9                )            
The parameters' physical meaning described allows the measurement of a linear N-port network, 2-ports at a time. Apply a stimulus on the port h, and measure the result on the port k. The ratio between those two quantities is the parameter of row index k and column index h. The difficulty of practically realizing that in the [Z]-case lies on the type of stimulus to apply. A current generator is needed, which is almost impossible to realize at RF frequencies. On the other side, the stimulus needed to measure the S-parameters is an incident wave generator, which means a generator with a finite output resistance equal or close to R0. Such generator, also known as matched generator, is much easier to realize at RF than the current generator. Thus, it follows that measuring the S-parameters is advantageous to measuring [Z] at high frequency. Furthermore, it follows that the minimum setup to measure a linear N-port network consists of one generator matched to R0, one receiver with input impedance also equal to R0, N−2 resistors with impedance Z=approx. R0, needed to terminate the N−2 other ports.
Thus, a conventional measurement technique first involves a matched generator (MG) which is a voltage generator with an output resistance equal or close to R0 and with known power, a matched receiver (MR) which is a voltage meter with an input resistance also equal or close to R0, wherein the measured voltage equals to the reflected wave, and N−2 resistors having a termination resistance at or close to R0 and which are connected between the remaining ports and ground to ensure the absence of incident waves at those ports. The MG additionally includes one receiver and one bridge or one directional coupler, to measure the wave reflected by the port stimulated by the MG.
Thus, a conventional method to measure S-parameters is to connect the MG to port 1 and the MR to port 2, so that the parameters s11 and s21 are measured while having the other ports terminated with a termination resistance at or close to R0. More in general, it comprises connecting the MG to port h and connecting the MR to port k. Then the parameters shk and shh are measured. Then the measurement may continue with the next couple of two ports. Thus two of the N×N S-parameters are measured at each measurement. The reflection parameter of each port is re-measured each time that port is excited. This means that (N−1)·(N−2) . . . =(N−1)! measurement steps may be used to complete the whole N×N S-parameters measurement. This brings the problem that many connections and disconnections may be used, although they can be automated by means of switching circuitry as known from multi-port network analyzers. At each measurement step, all the ports of the device to be tested (device under test, DUT) may be connected to something: the MG, or the MR, or the terminations. Thus, the conventional-technology method needs to simultaneously access all the N ports of the network to test. Sometimes the DUT is so small and the ports are so many and so close, that it is very difficult or simply not possible to access all of them simultaneously. Thus, there is a demand for an improved apparatus, an improved computer program, and an improved method for calculating one or more scattering parameters of a N-port linear network.